Digital Signal Processing - Classification of Discrete Time Signals

Just like Continuous time signals, Discrete time signals can be classified according to the conditions or operations on the signals.

Even and Odd Signals

Even Signal

A signal is said to be even or symmetric if it satisfies the following condition;

x(−n)=x(n)$$$$

Here, we can see that x(-1) = x(1), x(-2) = x(2) and x(-n) = x(n). Thus, it is an even signal.

Odd Signal

A signal is said to be odd if it satisfies the following condition;

x(−n)=−x(n)$$$$

From the figure, we can see that x(1) = -x(-1), x(2) = -x(2) and x(n) = -x(-n). Hence, it is an odd as well as anti-symmetric signal.

Periodic and Non-Periodic Signals

A discrete time signal is periodic if and only if, it satisfies the following condition −

x(n+N)=x(n)$$$$

Here, x(n) signal repeats itself after N period. This can be best understood by considering a cosine signal −

x(n)=Acos(2πf0n+θ)$$$$

x(n+N)=Acos(2πf0(n+N)+θ)=Acos(2πf0n+2πf0N+θ)

=Acos(2πf0n+2πf0N+θ)$$$$

For the signal to become periodic, following condition should be satisfied;

x(n+N)=x(n)

⇒Acos(2πf0n+2πf0N+θ)=Acos(2πf0n+θ)$$$$

i.e. 2πf0N$2\pi f_{0}N$ is an integral multiple of 2π$2\pi$

2πf0N=2πK

⇒N=Kf0

Frequencies of discrete sinusoidal signals are separated by integral multiple of 2π$2\pi$.

Energy and Power Signals

Energy Signal

Energy of a discrete time signal is denoted as E. Mathematically, it can be written as;

E=∑n=−∞+∞|x(n)|2$$$$

If each individual values of x(n) are squared and added, we get the energy signal. Here x(n) is the energy signal and its energy is finite over time i.e 0<E<∞

Power Signal

Average power of a discrete signal is represented as P. Mathematically, this can be written as;

P=limN→∞12N+1∑n=−N+N|x(n)|2$$$$

Here, power is finite i.e. 0<P<∞. However, there are some signals, which belong to neither energy nor power type signal.